Galois symmetries of fundamental groupoids

We give a simple proof of the formula for the coproduct ∆ in the Hopf algebra of motivic iterated integrals on the affine line. We show that it encodes the group law in the group of automorphisms of certain non commutative variety. We relate the coproduct ∆ with the coproduct in the Hopf algebra of decorated rooted planar trivalent trees defined in chapter 4 – a version of the one defined by Connes and Kreimer [CK]. As an application we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. 1. The Hopf algebra of motivic iterated integrals. Consider an iterated integral Iγ(a; z1, ..., zm; b) := ∫ a≤t1≤...≤tm≤b dt1 t1 − z1 ∧ dt2 t2 − z2 ∧ ... ∧ dtm tm − zm (1) Here γ is a path between the points a and b in C−{z1∪...∪zm}, and integration is over simplex consisting of all ordered m–tuples of points (t1, ..., tm) on γ. Iterated integral (1) is a period of a mixed Hodge structure or, better, mixed Tate motive. Therefore we can upgrade (1) to a more sophisticated object, the framed mixed Tate motive (see chapter 3 of [G3] for the background, and section